You have three doors, behind one is a car and behind the other two are goats. You are asked to pick one. What are the chances of you picking the car first time?

1 out of 3.

And what are the chances of you picking a goat?

2 out of 3.

So you are twice as likely to pick a goat as you are picking the car. Now the game show host steps up and opening one of the doors reveals a goat. There is now only one goat and the car left. So now what are the chances that you picked the car first time?

1 out of 3.

Just because you now know what is behind one of the doors doesn’t change the odds of you picking the car first time. You are still twice as likely to have picked a goat.

So knowing this we look at the last two doors. You know that you are twice as likely to have picked a goat first time. You also know where the other goat is. Now if you stick with your original choice your odds remain 1 out of 3 that you picked the car. However if you swap doors you also swap the odds. Because you had a 2 out of 3 chance of picking a goat by swapping the odds of correcting a mistake rather than making one are also 2 out of 3.

If you stayed with the first door you had a 1 out of 3 chance of being right.

By swapping you increase those odds to a 2 out of 3 chance of being right.

So swap.

Of course you could still loose but that’s what the games all about.